1. Field of the Invention
The present invention relates to a numerical stochastic integration method and system, and in particular to a technique by which stochastic integration of a Markov stochastic process can be calculated at high speed and with a small memory.
2. Related Art
Conventionally, “Single-factor Interest Rate Models And The Valuation Of Interest Rate Derivative Securities,” Hull J. and A. White, Journal of Financial and Quantitive Analysis, 28, 1993, pp. 235 to 254, is well known as a method whereby discrete approximation models for the Hull-White model and the BK model can be described by using a recombining trinomial tree structure by a computer. This method is further improved as is described in “Numerical Procedures For Implementing Term Structure Models I: Single-Factor Models,” Journal of Derivatives, 2, 1, Fall 1994, pp. 7 to 16. An example using this model is the “Hull-White Model Using EXCEL,” Nippon Credit Bank, Development Department, Financial Affairs Research Associates, 1996. The outline for a trinomial tree construction method will now be explained.
A discrete approximation model for the Hull-White model or the BK model is provided by acquiring a discrete approximation model used for a real valued diffusion process Xt, which is described by the following stochastic differential equation:dXt=−aXtdt+σdBt.  [Equation 1]
In this equation, t denotes time, Bt denotes the standard Brownian motion, and a denotes a positive constant. The constant a is called a central regression constant, and the term −aXtdt is called a central regression term. Since a Markov stochastic process is the stochastic process that is employed, discrete approximation can be performed by using a recombining trinomial tree structure shown in FIG. 1. In this case, the discrete width in the spatial direction is denoted by ΔX, and the discrete width along the time line is denoted by Δt. ΔX=σ√{square root over ( )}(3Δt) is established by analysis relative to a discretizing error that occurs from a Euler approximation. Each node in FIG. 1 corresponds to a state, and each arrow corresponds to an arrival probability (movement probability) at which the node is moved from a specific state to another state. It should be noted that in this process, unlike the normal Brownian motion, the spread of the discrete model in the spatial direction is halted en route because of the presence of the central regression term. The limiting point is represented by to in FIG. 1. In FIG. 1, the limits for this process are denoted by MΔX and −MΔX. In the Hull-White discrete model, the process can be successfully simplified so that tree structure branches at the above point can be described by using three patterns (a), (b) and (c) in FIG. 2.
Specifically, integer M in FIG. 1 is calculated in advance using [Equation 2]      0.184          a      ⁢                           ⁢      Δ      ⁢                           ⁢      t        <  M  <      0.186          a      ⁢                           ⁢      Δ      ⁢                           ⁢      t      
When the position of the node in the spatial direction reaches the upper limit, pattern (a) in FIG. 2 is employed; when the position of the node reaches the lower limit, pattern (c) is employed; and for the other cases, pattern (b) is employed. The arrival probability for each case can be easily calculated. Since when Δ−>0 the thus obtained discrete model converges in the stochastic process Xt, which is determined by the stochastic differential equation, the mathematically correct discrete process can be obtained.
It is known that a discrete process error due to the Euler approximation in the stochastic process is proportional to Δt. When a discrete interest rate model, such as the Hull-White model, is used for the calculation of the prices of derivatives, a very accurate discrete result of approximately Δt<{fraction (1/200)} is required in order to obtain a discrete process error that is not greater than 1 (bp), which is the practical ideal accuracy (see “Toward Real-time Pricing Of Complex Financial Derivatives,” Ninomiya, S., and S. Tezuka, Applied Mathematical Finance, 3, 1996, pp. 1 to 20). When the above trinomial tree structure for a ten-year interest period model is constructed that provide this accuracy, 500 MB or more of main memory is required, and the time required for the calculation of the price is unrealistic. Actually, since a 20 to 30 year time length is required for the interest period model in order to employ it to calculate the risk management probabilities for an asset, it is impossible for the conventional technique to perform the calculations with the required accuracy when a trinomial tree structure is used.